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Journal articles on the topic 'Euclidean algorithm'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Okazaki, Hiroyuki, Yosiki Aoki, and Yasunari Shidama. "Extended Euclidean Algorithm and CRT Algorithm." Formalized Mathematics 20, no. 2 (2012): 175–79. http://dx.doi.org/10.2478/v10037-012-0020-2.

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Summary In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm [9]. Besides the a gcd b, Extended Euclidean Algorithm can calculate a pair of two integers (x, y) that holds ax + by = a gcd b. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is to support the implementation of number theoretic tools. Our formalization of those algorithms is based on the source code of the NZMATH, a number theory oriented calculation system developed by

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2

Gilman, Jane. "The non-Euclidean Euclidean algorithm." Advances in Mathematics 250 (January 2014): 227–41. http://dx.doi.org/10.1016/j.aim.2013.09.012.

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3

Balkhair, Eynas. "Euclidean Algorithm Analysis." International Journal of Engineering Research and Applications 14, no. 12 (2024): 32–37. https://doi.org/10.9790/9622-14123237.

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4

Okazaki, Hiroyuki, Koh-ichi Nagao, and Yuichi Futa. "Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm." Formalized Mathematics 27, no. 1 (2019): 87–91. http://dx.doi.org/10.2478/forma-2019-0009.

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Summary In this article we formalize in Mizar [1], [2] the maximum number of steps taken by some number theoretical algorithms, “right–to–left binary algorithm” for modular exponentiation and “Euclidean algorithm” [5]. For any natural numbers a, b, n, “right–to–left binary algorithm” can calculate the natural number, see (Def. 2), AlgoBPow(a, n, m) := ab mod n and for any integers a, b, “Euclidean algorithm” can calculate the non negative integer gcd(a, b). We have not formalized computational complexity of algorithms yet, though we had already formalize the “Euclidean algorithm” in [7]. For “

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5

Afghani, Said Al, and Widhera Yoza Mahana Putra. "Clustering with Euclidean Distance, Manhattan - Distance, Mahalanobis - Euclidean Distance, and Chebyshev Distance with Their Accuracy." Indonesian Journal of Statistics and Its Applications 5, no. 2 (2021): 369–76. http://dx.doi.org/10.29244/ijsa.v5i2p369-376.

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There are several algorithms to solve many problems in grouping data. Grouping data is also known as clusterization, clustering takes advantage to solve some problems especially in business. In this note, we will modify the clustering algorithm based on distance principle which background of K-means algorithm (Euclidean distance). Manhattan, Mahalanobis-Euclidean, and Chebyshev distance will be used to modify the K-means algorithm. We compare the clustered result related to their accuracy, we got Mahalanobis - Euclidean distance gives the best accuracy on our experiment data, and some results

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6

Bras-Amorós, Maria, and Michael E. O’Sullivan. "The Symmetric Key Equation for Reed–Solomon Codes and a New Perspective on the Berlekamp–Massey Algorithm." Symmetry 11, no. 11 (2019): 1357. http://dx.doi.org/10.3390/sym11111357.

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This paper presents a new way to view the key equation for decoding Reed–Solomon codes that unites the two algorithms used in solving it—the Berlekamp–Massey algorithm and the Euclidean algorithm. A new key equation for Reed–Solomon codes is derived for simultaneous errors and erasures decoding using the symmetry between polynomials and their reciprocals as well as the symmetries between dual and primal codes. The new key equation is simpler since it involves only degree bounds rather than modular computations. We show how to solve it using the Euclidean algorithm. We then show that by reorgan

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7

MIERNOWSKI, TOMASZ, and ARNALDO NOGUEIRA. "Exactness of the Euclidean algorithm and of the Rauzy induction on the space of interval exchange transformations." Ergodic Theory and Dynamical Systems 33, no. 1 (2011): 221–46. http://dx.doi.org/10.1017/s014338571100085x.

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AbstractThe two-dimensional homogeneous Euclidean algorithm is the central motivation for the definition of the classical multidimensional continued fraction algorithms, such as Jacobi–Perron, Poincaré, Brun and Selmer algorithms. The Rauzy induction, a generalization of the Euclidean algorithm, is a key tool in the study of interval exchange transformations. Both maps are known to be dissipative and ergodic with respect to Lebesgue measure. Here we prove that they are exact.

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8

RUZSA, IMRE Z., and PETER P. VARJU. "Euclidean algorithm in different norms." Publicationes Mathematicae Debrecen 78, no. 1 (2011): 245–49. http://dx.doi.org/10.5486/pmd.2011.4804.

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9

Gorokhovskyi, Semen, and Artem Laiko. "Euclidean Algorithm for Sound Generation." NaUKMA Research Papers. Computer Science 4 (December 10, 2021): 48–51. http://dx.doi.org/10.18523/2617-3808.2021.4.48-51.

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Euclidean algorithm is known by humanity for more than two thousand years. During this period many applications for it were found, covering different disciplines and music is one of those. Such algorithm application in music first appeared in 2005 when researchers found a correlation between world music rhythm and the Euclidean algorithm result, defining Euclidean rhythms as the concept.In the modern world, music could be created using many approaches. The first one being the simple analogue, the analogue signal is just a sound wave that emitted due to vibration of a certain medium, the one th

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10

Van Den Dries, Lou, and Yiannis N. Moschovakis. "Is the Euclidean Algorithm Optimal Among its Peers?" Bulletin of Symbolic Logic 10, no. 3 (2004): 390–418. http://dx.doi.org/10.2178/bsl/1102022663.

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The Euclidean algorithm on the natural numbers ℕ = {0,1,…} can be specified succinctly by the recursive programwhere rem(a, b) is the remainder in the division of a by b, the unique natural number r such that for some natural number q,It is an algorithm from (relative to) the remainder function rem, meaning that in computing its time complexity function cε (a, b), we assume that the values rem(x, y) are provided on demand by some “oracle” in one “time unit”. It is easy to prove thatMuch more is known about cε(a, b), but this simple-to-prove upper bound suggests the proper formulation of the Eu

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11

Dong, Shu Juan. "Solving the Degree-Constrained Euclidean Steiner Minimal Tree Problem." Advanced Materials Research 219-220 (March 2011): 652–55. http://dx.doi.org/10.4028/www.scientific.net/amr.219-220.652.

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The degree-constrained Euclidean Steiner minimal tree problem was discussed based on the Euclidean Steiner minimal tree with each original point being added with a degree constraint. The property of the problem was analyzed and the implementation process of solving the problem by using the simulated annealing algorithm and the ant algorithm was presented. Both algorithms are coded in Delphi and run on the Windows XP environment. Series of numerical examples were tested and the efficiency of these algorithms was validated.

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12

Kalaidzhich, G. V. "Euclidean algorithm in matrix modules over a given Euclidean ring." Siberian Mathematical Journal 26, no. 6 (1986): 818–22. http://dx.doi.org/10.1007/bf00969102.

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13

Feng, Congcong, Bo Zhao, Xin Zhou, Xiaodong Ding, and Zheng Shan. "An Enhanced Quantum K-Nearest Neighbor Classification Algorithm Based on Polar Distance." Entropy 25, no. 1 (2023): 127. http://dx.doi.org/10.3390/e25010127.

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The K-nearest neighbor (KNN) algorithm is one of the most extensively used classification algorithms, while its high time complexity limits its performance in the era of big data. The quantum K-nearest neighbor (QKNN) algorithm can handle the above problem with satisfactory efficiency; however, its accuracy is sacrificed when directly applying the traditional similarity measure based on Euclidean distance. Inspired by the Polar coordinate system and the quantum property, this work proposes a new similarity measure to replace the Euclidean distance, which is defined as Polar distance. Polar dis

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14

GAVRILOVA, MARINA L., and MUHAMMAD H. ALSUWAIYEL. "TWO ALGORITHMS FOR COMPUTING THE EUCLIDEAN DISTANCE TRANSFORM." International Journal of Image and Graphics 01, no. 04 (2001): 635–45. http://dx.doi.org/10.1142/s0219467801000359.

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Given an n × n binary image of white and black pixels, we present two optimal algorithms for computing the distance transform and the nearest feature transform using the Euclidean metric. The first algorithm is a fast sequential algorithm that runs in linear time in the input size. The second is a parallel algorithm that runs in O(n2/p) time on a linear array of p processors, p, 1 ≤ p ≤ n.

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15

Chinta, Srujan Sai. "Kernelised Rough Sets Based Clustering Algorithms Fused With Firefly Algorithm for Image Segmentation." International Journal of Fuzzy System Applications 8, no. 4 (2019): 25–38. http://dx.doi.org/10.4018/ijfsa.2019100102.

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Data clustering methods have been used extensively for image segmentation in the past decade. In one of the author's previous works, this paper has established that combining the traditional clustering algorithms with a meta-heuristic like the Firefly Algorithm improves the stability of the output as well as the speed of convergence. It is well known now that the Euclidean distance as a measure of similarity has certain drawbacks and so in this paper we replace it with kernel functions for the study. In fact, the authors combined Rough Fuzzy C-Means (RFCM) and Rough Intuitionistic Fuzzy C-Mean

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16

Et.al, R. Felista Sugirtha Lizy. "Improvement of RSA Algorithm Using Euclidean Technique." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 3 (2021): 4694–700. http://dx.doi.org/10.17762/turcomat.v12i3.1889.

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Information Security has become an essential concern in the modern world. Encryption is an effective way to prevent an unofficial person from viewing the digital information with the secret key. RSA encryption is often used for digital signatures which can prove the authenticity and reliability of a message. As RSA encryption is less competent and resource-heavy, it is not used to encrypt the entire message. If a message is encrypted with a symmetric-key RSA encryption it will be more efficient. Under this process, only the RSA private key will be able to decrypt the symmetric key. The Euclide

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17

Rajasekaran, Sanguthevar. "On the Euclidean Minimum Spanning Tree Problem." Computing Letters 1, no. 1 (2005): 11–14. http://dx.doi.org/10.1163/1574040053326325.

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Given a weighted graph G(V;E), a minimum spanning tree for G can be obtained in linear time using a randomized algorithm or nearly linear time using a deterministic algorithm. Given n points in the plane, we can construct a graph with these points as nodes and an edge between every pair of nodes. The weight on any edge is the Euclidean distance between the two points. Finding a minimum spanning tree for this graph is known as the Euclidean minimum spanning tree problem (EMSTP). The minimum spanning tree algorithms alluded to before will run in time O(n2) (or nearly O(n2)) on this graph. In thi

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18

Peng, Xuesheng, Ruizhi Chen, Kegen Yu, Feng Ye, and Weixing Xue. "An Improved Weighted K-Nearest Neighbor Algorithm for Indoor Localization." Electronics 9, no. 12 (2020): 2117. http://dx.doi.org/10.3390/electronics9122117.

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The weighted K-nearest neighbor (WKNN) algorithm is the most commonly used algorithm for indoor localization. Traditional WKNN algorithms adopt received signal strength (RSS) spatial distance (usually Euclidean distance and Manhattan distance) to select reference points (RPs) for position determination. It may lead to inaccurate position estimation because the relationship of received signal strength and distance is exponential. To improve the position accuracy, this paper proposes an improved weighted K-nearest neighbor algorithm. The spatial distance and physical distance of RSS are used for

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19

Mwangi, Charles Irungu, and Moses Otieno Ouma. "Chairux Algorithm for Divisibility Test." Asian Journal of Advanced Research and Reports 17, no. 10 (2023): 111–41. http://dx.doi.org/10.9734/ajarr/2023/v17i10538.

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This paper presents a novel Chairux Algorithm for the divisibility test. The test is based on an arbitrary integer via the concepts of Bezout’s identity and the Euclidean algorithm. Examples demonstrating the effectiveness of the proposed algorithm indicate its simplicity, efficiency, and flexibility compared to the existing state-of-the-art algorithms.

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20

Sadiq, Afshan. "The F4-algorithm for Euclidean rings." Central European Journal of Mathematics 8, no. 6 (2010): 1156–59. http://dx.doi.org/10.2478/s11533-010-0064-x.

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21

Cook, Ian. "74.6 The Euclidean Algorithm and Fibonacci." Mathematical Gazette 74, no. 467 (1990): 47. http://dx.doi.org/10.2307/3618852.

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22

Harte, R. E. "Spectral Disjointness and the Euclidean Algorithm." Mathematical Proceedings of the Royal Irish Academy 118A, no. 2 (2018): 65–69. http://dx.doi.org/10.1353/mpr.2018.0008.

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23

Narkiewicz, Władysław. "Euclidean algorithm in small Abelian fields." Functiones et Approximatio Commentarii Mathematici 37, no. 2 (2007): 337–40. http://dx.doi.org/10.7169/facm/1229619657.

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24

Mujica, A. "Lattice reduction using a Euclidean algorithm." Acta Crystallographica Section A Foundations and Advances 73, no. 1 (2017): 61–68. http://dx.doi.org/10.1107/s2053273316015539.

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The need to reduce a periodic structure given in terms of a large supercell and associated lattice generators arises frequently in different fields of application of crystallography, in particular in theab initiotheoretical modelling of materials at the atomic scale. This paper considers the reduction of crystals and addresses the reduction associated with the existence of a commensurate translation that leaves the crystal invariant, providing a practical scheme for it. The reduction procedure hinges on a convenient integer factorization of the full period of the cycle (or grid) generated by t

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25

Lauritzen, Niels, and Jesper Funch Thomsen. "A Euclidean Algorithm for Integer Matrices." American Mathematical Monthly 126, no. 8 (2019): 699. http://dx.doi.org/10.1080/00029890.2019.1626667.

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26

Levrie, Paul, and Rudi Penne. "The extended Euclidean Algorithm made easy." Mathematical Gazette 100, no. 547 (2016): 147–49. http://dx.doi.org/10.1017/mag.2016.25.

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27

Lascoux, Alain, and Piotr Pragacz. "Bezoutians, Euclidean Algorithm, and Orthogonal Polynomials." Annals of Combinatorics 9, no. 3 (2005): 301–19. http://dx.doi.org/10.1007/s00026-005-0259-1.

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28

Antoulas, A. C. "Rational interpolation and the Euclidean algorithm." Linear Algebra and its Applications 108 (September 1988): 157–71. http://dx.doi.org/10.1016/0024-3795(88)90185-1.

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29

Kuijper, M. "Partial realization and the Euclidean algorithm." IEEE Transactions on Automatic Control 44, no. 5 (1999): 1013–16. http://dx.doi.org/10.1109/9.763219.

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30

Sunar, B. "A Euclidean Algorithm for Normal Bases." Acta Applicandae Mathematicae 93, no. 1-3 (2006): 57–74. http://dx.doi.org/10.1007/s10440-006-9048-z.

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31

R.E. Harte. "Spectral Disjointness and the Euclidean Algorithm." Mathematical Proceedings of the Royal Irish Academy 118A, no. 2 (2018): 65. http://dx.doi.org/10.3318/pria.2018.118.07.

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32

Embrechts, Hugo, and Dirk Roose. "A Parallel Euclidean Distance Transformation Algorithm." Computer Vision and Image Understanding 63, no. 1 (1996): 15–26. http://dx.doi.org/10.1006/cviu.1996.0002.

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33

Dong, Shu Juan, and Ai Ping Ding. "The Process of Ant Algorithm for Solving the Degree-Constrained Euclidean Steiner Minimal Tree." Advanced Materials Research 846-847 (November 2013): 1330–33. http://dx.doi.org/10.4028/www.scientific.net/amr.846-847.1330.

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The degree-constrained Euclidean Steiner minimal tree problem was discussed based on the Euclidean Steiner minimal tree with each original point being added with a degree constraint. The property of the problem was analyzed and the implementation process of solving the problem by using the ant algorithm was presented.The algorithms is coded in Delphi and run on the Windows XP environment.

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34

ZHOU, YAN, OLEKSANDR GRYGORASH, and THOMAS F. HAIN. "CLUSTERING WITH MINIMUM SPANNING TREES." International Journal on Artificial Intelligence Tools 20, no. 01 (2011): 139–77. http://dx.doi.org/10.1142/s0218213011000061.

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We propose two Euclidean minimum spanning tree based clustering algorithms — one a k-constrained, and the other an unconstrained algorithm. Our k-constrained clustering algorithm produces a k-partition of a set of points for any given k. The algorithm constructs a minimum spanning tree of a set of representative points and removes edges that satisfy a predefined criterion. The process is repeated until k clusters are produced. Our unconstrained clustering algorithm partitions a point set into a group of clusters by maximally reducing the overall standard deviation of the edges in the Euclidean

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35

Liu, Cong, Qianqian Chen, Yingxia Chen, and Jie Liu. "A Fast Multiobjective Fuzzy Clustering with Multimeasures Combination." Mathematical Problems in Engineering 2019 (January 17, 2019): 1–21. http://dx.doi.org/10.1155/2019/3821025.

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Most of the existing clustering algorithms are often based on Euclidean distance measure. However, only using Euclidean distance measure may not be sufficient enough to partition a dataset with different structures. Thus, it is necessary to combine multiple distance measures into clustering. However, the weights for different distance measures are hard to set. Accordingly, it appears natural to keep multiple distance measures separately and to optimize them simultaneously by applying a multiobjective optimization technique. Recently a new clustering algorithm called ‘multiobjective evolutionar

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36

Wang, Boyuan, Xuelin Liu, Baoguo Yu, Ruicai Jia, and Xingli Gan. "An Improved WiFi Positioning Method Based on Fingerprint Clustering and Signal Weighted Euclidean Distance." Sensors 19, no. 10 (2019): 2300. http://dx.doi.org/10.3390/s19102300.

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WiFi fingerprint positioning has been widely used in the indoor positioning field. The weighed K-nearest neighbor (WKNN) algorithm is one of the most widely used deterministic algorithms. The traditional WKNN algorithm uses Euclidean distance or Manhattan distance between the received signal strengths (RSS) as the distance measure to judge the physical distance between points. However, the relationship between the RSS and the physical distance is nonlinear, using the traditional Euclidean distance or Manhattan distance to measure the physical distance will lead to errors in positioning. In add

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37

Li, Ang, Peter J. Stuckey, Sven Koenig, and T. K. Satish Kumar. "Solving Facility Location Problems via FastMap and Locality Sensitive Hashing." Proceedings of the International Symposium on Combinatorial Search 17 (June 1, 2024): 46–54. http://dx.doi.org/10.1609/socs.v17i1.31541.

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Facility Location Problems (FLPs) arise while serving multiple customers in a shared environment, minimizing transportation and other costs. Hence, they involve the optimal placement of facilities. They are defined on graphs as well as in Euclidean spaces with or without obstacles; and they are typically NP-hard to solve optimally. There are many heuristic algorithms tailored to different kinds of FLPs. However, FLPs defined in Euclidean spaces without obstacles are the most amenable to efficient and effective heuristic algorithms. This motivates the idea of quickly reformulating FLPs on graph

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38

Wahyono, Wahyono, I. Nyoman Prayana Trisna, Sarah Lintang Sariwening, Muhammad Fajar, and Danur Wijayanto. "Comparison of distance measurement on k-nearest neighbour in textual data classification." Jurnal Teknologi dan Sistem Komputer 8, no. 1 (2019): 54–58. http://dx.doi.org/10.14710/jtsiskom.8.1.2020.54-58.

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One algorithm to classify textual data in automatic organizing of documents application is KNN, by changing word representations into vectors. The distance calculation in the KNN algorithm becomes essential in measuring the closeness between data elements. This study compares four distance calculations commonly used in KNN, namely Euclidean, Chebyshev, Manhattan, and Minkowski. The dataset used data from Youtube Eminem’s comments which contain 448 data. This study showed that Euclidian or Minkowski on the KNN algorithm achieved the best result compared to Chebycev and Manhattan. The best resul

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39

Luo, Zhikun, Huafei Sun, and Xiaomin Duan. "The Extended Hamiltonian Algorithm for the Solution of the Algebraic Riccati Equation." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/693659.

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We use a second-order learning algorithm for numerically solving a class of the algebraic Riccati equations. Specifically, the extended Hamiltonian algorithm based on manifold of positive definite symmetric matrices is provided. Furthermore, this algorithm is compared with the Euclidean gradient algorithm, the Riemannian gradient algorithm, and the new subspace iteration method. Simulation examples show that the convergence speed of the extended Hamiltonian algorithm is the fastest one among these algorithms.

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40

Erkan, Esra, and Salim Yüce. "Serret-Frenet Frame and Curvatures of Bézier Curves." Mathematics 6, no. 12 (2018): 321. http://dx.doi.org/10.3390/math6120321.

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The aim of this study is to view the role of Bézier curves in both the Euclidean plane E 2 and Euclidean space E 3 with the help of the fundamental algorithm which is commonly used in Computer Science and Applied Mathematics and without this algorithm. The Serret-Frenet elements of non-unit speed curves in the Euclidean plane E 2 and Euclidean space E 3 are given by Gray et al. in 2016. We used these formulas to find Serret-Frenet elements of planar Bézier curve at the end points and for every parameter t. Moreover, we reconstruct these elements for a planar Bézier curve, which is defined by t

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41

Han, Xiangyang, Senlin Wu, and Longzhen Zhang. "An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies." Axioms 13, no. 2 (2024): 132. http://dx.doi.org/10.3390/axioms13020132.

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In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global optimization algorithms having high computational complexity for this purpose. Since satisfactory estimations of covering functionals will be sufficient in Zong’s program, we propose a stochastic global optimization algorithm based on CUDA and provide an error estimation for the algorithm. The accuracy of our

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42

Wagon, Stan. "Editor's Corner: The Euclidean Algorithm Strikes Again." American Mathematical Monthly 97, no. 2 (1990): 125. http://dx.doi.org/10.2307/2323912.

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43

Oxman, Victor, and Avi Sigler. "Surprise Meeting: Euclidean Algorithm and Geometric Constructions." Resonance 27, no. 3 (2022): 435–42. http://dx.doi.org/10.1007/s12045-022-1331-4.

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44

Kim, Daehak, and Kwang Sik Oh. "Computer intensive method for extended Euclidean algorithm." Journal of the Korean Data and Information Science Society 25, no. 6 (2014): 1467–74. http://dx.doi.org/10.7465/jkdi.2014.25.6.1467.

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45

Miyazawa, M., Peifeng Zeng, N. Iso, and T. Hirata. "A systolic algorithm for Euclidean distance transform." IEEE Transactions on Pattern Analysis and Machine Intelligence 28, no. 7 (2006): 1127–34. http://dx.doi.org/10.1109/tpami.2006.133.

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46

Moore, Thomas E. "On the Least Absolute Remainder Euclidean Algorithm." Fibonacci Quarterly 30, no. 2 (1992): 161–65. http://dx.doi.org/10.1080/00150517.1992.12429372.

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47

Chastkofsky, Leonard. "The Subtractive Euclidean Algorithm and Fibonacci Numbers." Fibonacci Quarterly 39, no. 4 (2001): 320–23. http://dx.doi.org/10.1080/00150517.2001.12428711.

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48

Knopfmacher, Arnold. "Elementary Properties of the Subtractive Euclidean Algorithm." Fibonacci Quarterly 30, no. 1 (1992): 80–83. http://dx.doi.org/10.1080/00150517.1992.12429388.

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49

DANI, S. G., and ARNALDO NOGUEIRA. "On invariant measures of the Euclidean algorithm." Ergodic Theory and Dynamical Systems 27, no. 02 (2007): 417. http://dx.doi.org/10.1017/s0143385706000514.

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50

Weber, M., and Th M. Liebling. "Euclidean matching problems and the metropolis algorithm." Zeitschrift für Operations Research 30, no. 3 (1986): A85—A110. http://dx.doi.org/10.1007/bf01919172.

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